Two Approaches to Fractional Statistics in the Quantum Hall Effect: Idealizations and the Curious Case of the Anyon

Abstract

This paper looks at the nature of idealizations and representational structures appealed to in the context of the fractional quantum Hall effect, specifically, with respect to the emergence of anyons and fractional statistics. Drawing on an analogy with the Aharonov–Bohm effect, it is suggested that the standard approach to the effects—(what we may call) the topological approach to fractional statistics—relies essentially on problematic idealizations that need to be revised in order for the theory to be explanatory. An alternative geometric approach is outlined and endorsed. Roles for idealizations in science, as well as consequences for the debate revolving around so-called essential idealizations, are discussed.

Notes

Acknowledgments

Parts of this paper were presented at the “Bucharest Colloquium in Analytic Philosophy: New Directions in the Philosophy of Physics” conference at University of Bucharest. I am grateful to the audience for stimulating discussion, to Iulian Toader for editing this volume of Foundations of Physics, and to two anonymous referees for helpful comments. I am especially grateful to John Earman and John D. Norton for comments on earlier versions of this paper, numerous insightful discussions, and their constant support. This paper is heavily in debt to work done by John Earman, who initially drew my attention to the main issues discussed in this paper, and has been especially kind in helping me work through the finer details. Special thanks to Naharin Shech for her help with figures.

Where \(Z\) is the cyclic group of order one, i.e., the infinite group of integers under addition. \(Z_2 \) is the cyclic group of order two, i.e., it is the multiplicative group of, say, 1 and \(-\)1. \(RP_1\) and \(RP_2 \) are the real projective one- and two-dimensional spaces, respectively.

Pictorially, for the \(d=3\) case the configuration space reduces to the real projective space in two dimensions \(RP_2\). This can be visualized as the surface of a three-dimensional sphere with diametrically opposite points identified (see Fig. 13). Consider three scenarios, corresponding to three paths \(A\), \(B\), and \(C\) in configuration space including no exchange (Fig. 13a), exchange (Fig. 13b), and a double exchange (Fig. 13c), respectively.

The real projective space in two dimensions \(RP_2 \), represented by a sphere with diametrically opposite points identified. Cases (a), (b), and (c) correspond to no exchange, exchange, and double exchange, respectively

Concentrating on the no exchange case (Fig. 13a). We trace a path \(A\) in configuration space in which the two particles move and return to their original positions. Path \(A\) is a loop in configuration space, with the same fixed start and end points, which can be shrunk to a point. This correspond to a trivial homotopy class in which the phase factor is trivial.

Moving onto the exchange case (Fig. 13b), we start at one end of the configuration space and trace a path \(B\) to its diametrically opposite point. This represents an exchange or permutation between the two particles. Notice that since diametrically opposite points are identified (because the particles are identical), this path is actually a closed loop in configuration space. However, since the start and end points of Fig. 15b are fixed, the loop cannot be shrunk to point. This corresponds to a non-trivial homotopy class with a non-trivial phase factor.

The double exchange (Fig. 13c) case includes tracing a path \(C\) in configuration space similar to that of \(B\), but then tracing around the sphere back to the original starting point. Path \(C\) is a closed loop in configuration space that can be shrunk to a point, and so it is in the same homotopy class of path \(A\) with a corresponding trivial phase factor. Equivalently, we may visualize the paths \(A\), \(B\), \(C\) on a hemisphere with opposite points on the equator identified as in Fig.14, where paths \(A\) and \(C\) can be continuously deformed to a point but path B cannot because of the diametrically opposed fixed start and end point on the equator.

The real projective space in two dimensions \(RP_2 \), represented by the northern hemisphere with opposite point on the equator identified

On the other hand, in the context of the \(d=2\) case, we are dealing with the real projective space in one dimension \(RP_1 \). We can visualize this configuration space as a circle with diametrically opposite points identified (see Fig. 15). Again, consider three paths \(A\), \(B\), and \(C\) in configuration space that correspond to no exchange (Fig. 15a), exchange (Fig. 15b), and a double exchange (Fig. 15c), respectively. Path \(A\) traces a closed loop in configuration space (where the particles move but then return to their original positions with no exchange) which can be continuously shrunk to a point and has a corresponding trivial phase factor (as in the \(d=3\) case of Fig. 14a). Next, we trace a path \(B\) across half the circumference of the circle. Since diametrically opposed points are identified, this represents a particle exchange (Fig. 15b). Path B traces a closed loop in configuration space that cannot be continuously shrunk to a point and has a corresponding non-trivial phase factor (as in the \(d=3\) case of Fig. 14b).

The real projective space in one dimension \(RP_1\), represented by a circle with diametrically opposite points identified. Cases (a), (b), and (c) correspond to no exchange, exchange, and double exchange, respectively

The main difference between the \(d=3\) and \(d=2\) cases arises when we consider path \(C\) (Fig. 15c), in which the particles are permuted twice, represented by traversing the entire circular configuration space. Path \(C\) is a closed loop in configuration space but, unlike the \(d=3\) case, it cannot be shrunk to a point because the circle itself (so to speak) acts as an obstructive barrier. Moreover, path \(C\) cannot even be continuously deformed to overlap with path B. This means that, not only is the phase factor corresponding to the two paths non-trivial, but each path has a different phase factor for each path belongs to a different homotopy class. In fact, for every traversal (in configuration space) of half a circle, we get a closed loop that is in its own homotopy class.47 In other words, by transitioning from three dimensions to two dimensions, we have transitioned from a doubly connected space to an infinitely connected space, and it is this change in topology that allows for intermediate statistics.

Appendix 2: A Geometric Approach to Fractional Statistics

We follow Berry’s [21] original paper on the non-dynamical phase factor accompanying cyclic evolutions of quantum systems. To begin, consider some system, such as a spinless electrically charged particle in a box, with a corresponding Hamiltonian \(H( {\varvec{R}( t)})\) that depends on a set of parameters \(\varvec{R}=(X,Y,\ldots )\) and can be altered over time by varying said parameters. We can view the alteration of \(H( {\varvec{R}( t)})\) as a path in parameter space. If the system starts at some time \(t=0\), and is gradually changed over time \(t\) so that the parameter values are returned to their original values \(\varvec{R}(0)=\varvec{R}(t)\) then this maps out a closed curve \(C\) in parameter space. According to the adiabatic theorem, if the system was originally (at time \(t=0)\) in the \(n\)th eigenstate \({\psi }_n (\varvec{R}(0))\) of \(H( {\varvec{R}( 0)})\), if \(H( {\varvec{R}( t)})\) is non-degenerate, and if the excursion in parameter space is sufficiently slow, then the system will transition (under Schrödinger evolution) into the \(n\)th eigenstate of \({\psi }_n ( {\varvec{R}(t)})\) of \(H( {\varvec{R}( t)})\) (with some added phase factor).48

The general state of the system \({\Psi }( t)\) evolves according to the time-dependent Schrödinger equation, and atany instant\(t\) the eigenstates of the time-independent Schrödinger equation form a natural basis satisfying:

\(\theta _D \) corresponds to the usual dynamical phase (accompanying the Schrödinger evolution of any stationary state) and \(\theta _G \)is called the geometric phase or Berry’s phase (where I have used Dirac’s bra-ket notation and hid the parameter dependence for convenience). It can be expressed more generally as a quantity dependent on both the closed curve \(C\) in parameter space and the parameters \({\varvec{R}}=(X,Y,\ldots )\):

Where \(\nabla _R \) is the gradient with respect to the parameters \({\varvec{R}}=(X,Y,\ldots )\) (and assuming that \({\varvec{R}}( 0)=R({\varvec{t}})\) so that \(C\) forms a closed curve). There are similar results for degenerate systems [110] and for a cyclic evolution that is not necessarily adiabatic [1].

Next, my goal in the rest of this appendix is solely to repeat some of the steps taken by Arovas, Schrieffer, and Wilczek [7] to derive fractional statistics in order to emphasize the disconnect between this geometric approach and the topological (and pathological) approach discussed in Sect. 4. Many steps will be skipped, and I refer the reader interested in a more details to explicit calculations made by Arovas in [95], pp. 284–322] and Laughlin in [109], pp. 262–303].

Following [6, 7, 64, 65, 66] closely, let us consider a FQHE system with filling factor\(\nu =\frac{1}{m}\) where \(m\) is an odd integer, and the applied strong magnetic field \(B\) is in the \(z\)-axis direction corresponding to magnetic flux \({\Phi }.\) In such a situation, the Hamiltonian governing the system is50:

Recall, \(z_j \equiv x_j +iy_i \) are in units of magnetic length \(l_B \equiv \sqrt{\hbar c/eB} \), which have been set to equal one, \(e\) is the charge of the electron, and \(j\) and \(k\) run over \(N\) particles. The \(\frac{( {p_j -qA_j })^2}{2m}\) term signifies the kinetic energy of charged particles in a magnetic field, \(\varvec{V}(z_j )\) is average background potential, and \(\frac{e^2}{\left| {z_j -z_k } \right| }\) is the Coulomb interaction between particles. Laughlin’s [64, 65] celebrated wavefunction for the ground state of \(H_{FQHE}^I \) is:

We can determine the quantum statistics associated with exchanging quasiholes \(a\) and \(b\) by calculating the geometric phase associated with carrying quasihole \(a\) adiabatically around a closed loop \(C\), thereby adding time dependence to \(z_a =z_a ( t),\) and identifying the geometric phase with the exchange phase. The geometric phase \(\theta _G \) can be calculated by plugging Eq. 7.3 into Eq. 7.1 or 7.2 as follows:

Denoting the mean number of electrons inside loop \(C\) with\(\langle {n_{e}}\rangle _C \), it turns out that solving Eq. 7.4 leads to the following expression for the geometric phase \(\theta _G =-2\pi \langle {n_{e}}\rangle _C \).51 If quasihole \(b\) is outside the loop then \(\langle {n_{e}}\rangle _C \) is equal to \(\frac{\nu {\Phi }}{{\Phi }_0 }\), where \(\nu \) is the filling factor, \({\Phi }\) is the magnetic flux corresponding to the strong magnetic field applied in FQHE systems, and the constant \({\Phi }_0 \equiv \frac{hc}{e}\) is the “flux quanta,” so that \(\theta _G =-2\pi \frac{\nu {\Phi }}{{\Phi }_0 }\). However, if quasihole \(b\) is inside the loop then there is a deficit in mean number of electrons by an amount –\(\nu \) so that \(\theta _G =-2\pi \frac{\nu {\Phi }}{{\Phi }_0 }+2\pi \nu \). The difference in geometric phase between the two scenarios is \({\Delta }\theta _G =2\pi \nu \).

Where we have introduced the “statistical parameter” defined as \(\alpha \equiv \frac{\theta }{\pi }\). We see that \(\alpha =\nu \) and recalling that \(\nu =\frac{1}{m}\) where \(m\) is an odd integer, it follows that \(\theta =\frac{\pi }{m}\) . For the m=1 case, \(\theta =\pi \) corresponding to Fermi–Dirac statistics. But for other values of \(m\), \(\theta \) corresponds to anyonic statistics.